Chapter 3

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=3 x^{2}-x^{3}$$

The width of a rectangular box is twice the height and the length is 7 inches more than the height. If the volume is 72 cubic inches, find the dimensions of the box. (GRAPH CANNOT COPY).

Explain how to perform long division of polynomials. Use \(2 x^{3}-3 x^{2}-11 x+7\) divided by \(x-3\) in your explanation.

You have 120 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=\frac{1}{2}-\frac{1}{2} x^{4}$$

Use the four-step procedure for solving variation problems given on page 356 to solve. The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. A wire of 720 feet with \(\frac{1}{4}\)-inch diameter has a resistance of \(1 \frac{1}{2}\) ohms. Find the resistance for 960 feet of the same kind of wire if its diameter is doubled.

The United States has more people in prison, as well as more people in prison per capita, than any other western industrialized nation. The bar graph shows the number of inmates in U.S. state and federal prisons in seven selected years from 1985 through 2000 (GRAPH CANNOT BE COPY) The data in the graph can be modeled by a linear function, \(f(x)=61.3 x+495\) a quadratic function, \(g(x)=-0.131 x^{2}+63.27 x+491.6\) a third-degree polynomial function, \(h(x)=-0.219 x^{3}+4.885 x^{2}+35.14 x+503.14\) For each of these functions, \(x\) represents the number of years after 1985 and the function value represents the number of inmates, in thousands. Use this information to solve Exercises \(48-49\). The graph indicates that in \(2000,\) there were 1382 thousand inmates. Substitute 1382 for \(f(x)\) and \(g(x)\) in the linear and quadratic models. Then solve each resulting equation to find how many years after \(1985,\) to the nearest tenth of a year, inmate population was 1382 thousand. How well do the linear and quadratic functions serve as a model for \(2000 ?\)

A box with an open top is formed by cutting squares out of the corners of a rectangular piece of cardboard 10 inches by 8 inches and then folding up the sides. If \(x\) represents the length of the side of the square cut from each corner of the rectangle, what size square must be cut if the volume of the box is to be 48 cubic inches? (GRAPH CANNOT COPY).

In your own words, state the Division Algorithm.

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-3(x-1)^{2}\left(x^{2}-4\right)$$

The United States has more people in prison, as well as more people in prison per capita, than any other western industrialized nation. The bar graph shows the number of inmates in U.S. state and federal prisons in seven selected years from 1985 through 2000 (GRAPH CANNOT BE COPY) The data in the graph can be modeled by a linear function, \(f(x)=61.3 x+495\) a quadratic function, \(g(x)=-0.131 x^{2}+63.27 x+491.6\) a third-degree polynomial function, \(h(x)=-0.219 x^{3}+4.885 x^{2}+35.14 x+503.14\) For each of these functions, \(x\) represents the number of years after 1985 and the function value represents the number of inmates, in thousands. Use this information to solve Exercises \(48-49\). The graph indicates that in \(2000,\) there were 1382 thousand inmates. Substitute 1382 for \(h(x)\) in the third degree model. Set the resulting equation equal to 0 and show that it has a real root between 14 and \(15 .\) Then use the Intermediate Value Theorem or a graphing utility's zero feature to find an approximation, to the nearest tenth, for this root. How well does the third-degree polynomial function serve as a model for \(2000 ?\)

Describe how to find the possible rational zeros of a polynomial function.

How can the Division Algorithm be used to check the quotient and remainder in a long division problem?

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=-2(x-4)^{2}\left(x^{2}-25\right)$$

What does it mean if two quantities vary directly?

When testing a number using synthetic division, how do you know if it is an upper bound for the real roots?

Describe how to use Descartes's Rule of Signs to determine the possible number of positive real zeros of a polynomial function.

Hunky Beef, a local sandwich store, has a fixed weekly cost of \(\$ 525.00,\) and variable costs for making a roast beef sandwich are \(\$ 0.55\) a. Let \(x\) represent the number of roast beef sandwiches made and sold each week. Write the weekly cost function, \(C,\) for Hunky Beef. (Hint: The cost function is the sum of fixed and variable costs.) b. The function \(R(x)=-0.001 x^{2}+3 x\) describes the money, in dollars, that Hunky Beef takes in each week from the sale of \(x\) roast beef sandwiches. Use this revenue function and the cost function from part (a) to write the stores weekly profit function, \(P\). (Hint: The profit function is the difference between revenue and cost functions.) c. Use the stores profit function to determine the number of roast beef sandwiches it should make and sell each week to maximize profit. What is the maximum weekly profit?

A herd of 100 elk is introduced to a small island. The number of elk, \(N(t),\) after \(t\) years is described by the polynomial function \(N(t)=-t^{4}+21 t^{2}+100\) a. Use the Leading Coefficient Test to determine the graphs end behavior to the right. What does this mean about what will eventually happen to the elk population? b. Graph the function. c. Graph only the portion of the function that serves as a realistic model for the elk population over time. When does the population become extinct?

In your own words, explain how to solve a variation problem.

Describe how to use Descartes's Rule of Signs to determine the possible number of negative roots of a polynomial equation.

State the Remainder Theorem.

What is a quadratic function?

The common cold is caused by a rhinovirus. After \(x\) days of invasion by the viral particles, the number of particles in our bodies, \(f(x),\) in billions, can be modeled by the polynomial function $$ f(x)=-0.75 x^{4}+3 x^{3}+5 $$ Use the Leading Coefficient Test to determine the graphs end behavior to the right. What does this mean about the number of viral particles in our bodies over time?

What does it mean if two quantities vary inversely?

How do you show that a polynomial function has a real zero between two given numbers?

Why must every polynomial equation of degree 3 have at least one real root?

Explain how the Remainder Theorem can be used to find \(f(-6)\) if \(f(x)=x^{4}+7 x^{3}+8 x^{2}+11 x+5 .\) What advantage is there to using the Remainder Theorem in this situation rather than evaluating \(f(-6)\) directly?

What is a parabola? Describe its shape.

The polynomial function $$ f(x)=-0.87 x^{3}+0.35 x^{2}+81.62 x+7684.94 $$ models the number of thefts, \(f(x),\) in thousands, in the United States \(x\) years after \(1987 .\) Will this function be useful in modeling the number of thefts over an extended period of time? Explain your answer.

Explain what is meant by combined variation. Give an example with your explanation.

How does the linear factorization of \(f(x),\) that is, $$f(x)=a_{n}\left(x-c_{1}\right)\left(x-c_{2}\right) \cdots\left(x-c_{n}\right)$$ show that a polynomial equation of degree \(n\) has \(n\) roots?

Explain why the equation \(x^{4}+6 x^{2}+2=0\) has no rational roots.

How can the Factor Theorem be used to determine if \(x-1\) is a factor of \(x^{3}-2 x^{2}-11 x+12 ?\)

Explain how to decide whether a parabola opens upward or downward.

Explain what is meant by joint variation. Give an example with your explanation.

Show that \(-1\) is a lower bound of \(f(x)=x^{3}-53 x^{2}+\) \(103 x-51 .\) Show that 60 is an upper bound. Use this information and a graphing utility to draw a relatively complete graph of \(f\).

Suppose \(\frac{3}{4}\) is a root of a polynomial equation. What does this tell us about the leading coefficient and the constant term in the equation?

If you know that \(-2\) is a zero of $$f(x)=x^{3}+7 x^{2}+4 x-12$$ explain how to solve the equation $$x^{3}+7 x^{2}+4 x-12=0$$

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

Describe in words the variation shown by the given equation. \(z=\frac{k \sqrt{x}}{y^{2}}\)

In Exercises \(55-56,\) use a graphing utility to determine upper and lower bounds for the zeros of \(f .\) Does synthetic division verify your observations? $$ f(x)=2 x^{3}+x^{2}-14 x-7 $$

The idea of supply-side economics (see Exercises 45-46 ) is that an increase in the tax rate may actually reduce government revenue. What explanation can you offer for this theory?

Describe how to find a parabola's vertex if its equation is in the form \(f(x)=a x^{2}+b x+c\). Use \(f(x)=\) \(x^{2}-6 x+8\) as an example.

What is a polynomial function?

Describe in words the variation shown by the given equation. \(z=k x^{2} \sqrt{y}\)

In Exercises \(55-56,\) use a graphing utility to determine upper and lower bounds for the zeros of \(f .\) Does synthetic division verify your observations? $$ f(x)=2 x^{4}-7 x^{3}-5 x^{2}+28 x-12 $$

The equations have real roots that are rational. Use the Rational Zero Theorem to list all possible rational roots. Then graph the polynomial function in the given viewing rectangle to determine which possible rational roots are actual roots of the equation. $$ 2 x^{3}-15 x^{2}+22 x+15=0 ;[-1,6,1] \text { by }[-50,50,1] $$

Use a graphing utility to determine if the division has been performed correctly Graph the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side by using polynomial long division. Then verify your correction using the graphing utility. $$\left(6 x^{2}+16 x+8\right) \div(3 x+2)=2 x+4, x \neq-\frac{2}{3}$$

What do we mean when we describe the graph of a polynomial function as smooth and continuous?